The functional minimization of the KL divergence in (REF ) is convex with respect to a single variational pdf \(Q(\Theta _j)\)  when the others \(\lbrace Q(\Theta _j)\rbrace _{i\ne j}\)  are fixed. This guarantees monotonic decrease of the KL divergence in (REF ) over iterations. In addition, the proposed CTAD algorithm is an instance of the block coordinate descent optimization strategy over the functional space. Therefore, the CTAD algorithm is able to decrease the objective in every iteration. Since the proposed algorithm is developed under the mean-field variational inference framework [1]}, the limit point of the proposed algorithm is guaranteed to converge to a stationary point of the objective function in (REF ).
